UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI
2003
THE CLASSIFICATION OF TILING SPACE FLOWS
by Alex Clark
Abstract. We consider the conjugacy of the natural flows on one-dimensional tiling spaces presented as inverse limits. We also draw connections
between geometric models and the spectral information for such flows.
1. Introduction. Our goal here is to present some of the results on classifying the flows on one-dimensional substitution tiling spaces in [8] from the
perspective of inverse limits, to emphasize the features of those results that
follow from this perspective, to extend some of those results to more general
tiling spaces, and to demonstrate how to provide a geometric model of the
tiling space when it has pure point spectrum.
If P = {P1 , ..., Pn } is a collection of intervals (prototiles), then a tiling T
of R based on P is a collection of intervals (tiles) {Ti }i∈Z satisfying:
1. Each Ti a translate of some Pj ∈ P
2. ∪i∈Z Ti = R
3. Ti ∩ Ti+1 is a singleton for each i.
There is a metric on T (P) , the tilings of R based on P, by which two
tilings T and T 0 are close if there is a small ε > 0 so that the tiles of T and T 0
in a large neighborhood of 0 agree up to translation by some number < ε [1].
Given any T = {Ti }i∈Z ∈ T (P) and t ∈ R, T − t = {Ti − t}i∈Z ∈ T (P) , and
so there is the natural continuous flow
φ
φ : R × T (P) → T (P) ; (t, T ) 7→ T − t
that moves the origin of a tiling t units forward along the tiling after t units
of time. Given a particular T ∈ T (P) , the tiling space T of T is the closure
of the φ− orbit of T. The restriction φ|T is then the natural flow on T .
1991 Mathematics Subject Classification. Primary 52C23; Secondary 37A25, 37A30,
37A10, 37B10.
Key words and phrases. Tiling space, flow, geometric model.
This research was funded in part by a grant from the University of North Texas.
50
We begin by considering tiling spaces presented as an inverse limit space
f0
f1
K0 ← K1 ← K2 · · · lim {Ki ; fi } = T ,
)
where each Ki is a PL wedge of n of circles and where each fi is a PL local isometry, represented by the integral matrix (using row multiplication)
Mi : Zn → Zn giving the homomorphism (fi )∗ : H1 (Ki+1 ) → H1 (Ki ) . We construct the inverse limit representation of T to reflect the natural flow structure.
If T ∈ T (P = {P1 , ..., Pn }) , then K0 is the wedge of n circles K01 , ..., K0n with
the circumference of K0i the length of Pi . If ρ0 (x, y) denotes the minimum
length of any arc joining the two points x, y ∈ K0 , then as a metric for K0 we
use
d0 (x, y) = min {L1 , . . . , Ln , ρ0 (x, y) , 1} ,
where Lj is the length of K0j . Then for all i > 0 the circumferences of the
circles Ki1 , ..., Kin are determined by requiring the PL bonding maps fj to be
local isometries. This in turn determines metrics ρi and di by analogy. We
then define a metric d for T by:
∞
X
1
∞
d (hxi i∞
,
hy
i
)
=
di (xi , yi ) .
i i=0
i=0
i+1
2
i=0
Then if pi : T → Ki denotes the projection, the natural flow φ on T projects
to a branched flow on Ki , which is well-defined and locally isometric except
at the branch point. We orient the circles in Ki to coincide with the direction
of the flow. Moreover, if y = φ (t, x) , then di (xi , yi ) ≤ t for all i and so
d (x, y) ≤ t. We shall examine when two natural flows φ and ψ are conjugate
for homeomorphic tiling spaces T and S with different choices of tile lengths.
2. Sufficient conditions for Conjugacy. In this section we provide
manageable conditions for conjugacy. We first treat the “substitution” case
with Mk ≡ M and L= (L1 , ..., Ln ) , S= (S1 , ..., Sn ) the circumferences of the
circles wedged to form K0 and J0 used in the construction of T and S respectively. Here we are assuming that the bonding maps fi for T and gi for S
determine the same association of circles, and so only differ in the lengths of
the circles in the domain and range spaces.
Theorem 1. The natural flows on T and S are conjugate if there exists
an integer k so that
lim LM i+k − SM i = (0, . . . , 0) .
i→∞
Proof. We first show that any two such flows meeting the condition for
k = 0 are conjugate. To start, we construct a homeomorphism h0 : T → S
induced by the PL homeomorphism λ00 : K0 → J0 which maps K0j linearly and
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orientation preserving onto J0j , thereby determining a sequence of homeomorphisms λj0 : Kj → Jj making the following diagram and its vertical inverse
commute
f0
f1
K0 ← K1 ← K 2 · · ·
↓ λ00
↓ λ10
↓ λ20
g0
g1
J0 ← J1 ← J2 · · ·
T
↓ h0 .
S
For each i = 1, 2, ... there is an analogous PL homeomorphism λii : Ki → Ji
which maps each Kij linearly and orientation preserving onto Jij , but this homeomorphism does not lead to a complete diagram as before since there are no
well-defined commuting vertical maps for k < i. However, as the homeomorphism type of an inverse limit is unchanged by dropping off any finite number
of initial factors in the defining sequence, the commutative diagram
fi
Ki ← Ki+1
↓ λii
↓ λi+1
i
gi
Ji ← Ji+1
fi+1
←
gi+1
←
Ki+2 · · ·
↓ λi+2
i
Ji+2 · · ·
T
↓ hi
S
induces a homeomorphism hi : T → S. The homeomorphism hi identifies the
supertiles of order i. Moreover, each hi induces the same correspondence of
path components, moving points to varying places in the same flow orbit.
We now proceed to show that {hi } forms a Cauchy sequence of homeomorphisms in the space of homeomorphisms T → S in the sup metric D. The
vectors LM i and SM i give the circumferences of the circles in Ki and Ji . Since
we have constructed the metrics to locally correspond to length in Ki and Ji ,
we then see that the map λii distorts length and hence distance by at most µi ,
the maximum difference in the entries of LM i and SM i . Moreover, comparing
the construction of hi with that of hi+k , we see that
D (hi , hi+k ) ≤
i
X
µi + µi+k
`=0
2`+1
+
i
X
`=i+1
1
2`+1
< µi + µi+k +
1
.
2i
Hence, {hi } is a Cauchy sequence of homeomorphisms. As T and S are compact metric spaces, the space of homeomorphisms T → S is complete relative
to D, and so {hi } converges to a homeomorphism h : T → S, which then
conjugates the natural flows on T and S since the {hi } preserve time up to
{µi } → 0 over supertiles of order i.
Now assume that the condition is met for some k > 0. The k th iterate of
the shift map of T conjugates the natural flow on T with the natural flow on
f
fk+1
k
Kk ←
Kk+1 ← Kk+2 · · · lim {Ki ; fi }i≥k = T 0 ,
)
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where the circumferences of the circles in Kk are given by LM k . Then the
natural flows on T 0 and S are conjugate by the k = 0 case. The case k < 0
can be handled similarly to construct a conjugacy S → T .
In [8] a substitution tiling space flow is the special flow (suspension) under
a function f of a substitution subshift on a finite alphabet, {a1 , ..., an }Z , where
the function f depends only on the letter corresponding to 0 ∈ Z. The above
result does not apply directly to all the substitution tiling spaces considered
in [8], but as the following shows, the above result can be applied to this more
general setting.
Corollary 1. If S and T are one-dimensional substitution tiling spaces
generated by the same substitution but with tile lengths given by L and S respectively, then the natural flows on T and S are conjugate if there exists an
integer k so that
lim LM i+k − SM i = (0, . . . , 0) ,
i→∞
where the matrix M represents the substitution.
Proof. Let Ki be the supertiles in T of order i wedged at a single point,
with lengths given by LM i . Let fi : Ki+1 → Ki be the map determined by
the substitution, essentially what is referred to in [3] as the map of the rose
(only here we vary lengths in the Ki ). The inverse limit
f0
f1
K 0 ← K1 ← K2 · · ·
T0
is not homeomorphic to T unless the original substitution is proper (forces
the border), see [1], [3]. Consider, however, the natural mapping p : T → T 0 ,
p (T ) = hpi (T )i∞
i=0 , where pi (T ) assigns to the tiling T the position of the
origin in T within its ith order supertile, which is well defined by the results of
[12], [13]. Then two tilings T, T 0 have the same p value only if the origins of T
and T 0 are in the same position relative to all order supertiles, which can only
happen if the flow orbits of T and T 0 are asymptotic in either the forward or
backward time direction, as mentioned in [3]. It then follows that the mapping
p respects the time structure of the flow and identifies at most finitely many
asymptotic flow orbits. Similarly we construct S 0 and q : S → S 0 . Under the
stated condition, we can then construct a length preserving homeomorphism
h0 : T 0 → S 0 as before. What is more, h0 associates the path components in
T 0 and S 0 which are images of more than one orbit from the original spaces.
Thus, the map h0 lifts to a homeomorphism h : T → S which maps the orbits
identified in T 0 and S 0 as determined by h0 . Since p and q preserve length along
orbits, it then follows that h is a conjugacy.
Substitutions having a Pisot matrix representation have been well studied,
see, e.g., [2]. Any matrix of Pisot type is diagonalizable and has a single
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eigenvalue of modulus greater than one. Thus, if M is an n × n matrix of Pisot
type with dominant eigenvector v1 , then the natural flows corresponding to
(L1 , L2 , ..., Ln ) = a1 v1 + · · · + an vn and (S1 , S2 , ..., Sn ) = b1 v1 + + · · · + bn vn
are conjugate if a1 = b1 , where {vi } is a basis of left eigenvectors. In fact,
all that is necessary to conclude conjugacy up to a linear rescaling is that the
only eigenvalue of M that has modulus 1 or greater is the Perron eigenvalue.
Hence, up to a time-scale factor any two such flows are conjugate, generalizing
the result of [14] for the Fibonacci substitution.
If the substitution is an invertible substitution on two letters, then the
inverse limit space is homeomorphic to a suspension of a Sturmian subshift for
some quadratic irrational α [2]; in other contexts such a space is referred to
as a Denjoy continua [4]. Any such Sturmian subshift has discrete spectrum,
and since the matrix of such a substitution (being unimodular) will have one
eigenvalue larger and one eigenvalue smaller than 1 in absolute value, the above
results imply that the natural flow on any such tiling space has pure discrete
spectrum.
We now treat the general (not necessarily substitution) case (M1 , M2 , . . . )
with corresponding bonding maps (f1 , f2 , . . . ) .
Theorem 2. The natural flows on
f1
f2
K0 ← K1 ← K2 · · ·
T
∼ L = (L1 , . . . , Ln )
and
g1
g2
J0 ← J1 ← J2 · · ·
S ∼ S = (S1 , . . . , Sn )
are conjugate if
lim (LM1 · · · Mi − SM1 · · · Mi ) = (0, . . . , 0) .
i→∞
Proof. Just as in the k = 0 case of Theorem 1, construct a Cauchy
sequence of homeomorphisms {hi } converging to a conjugacy.
The Denjoy continua topologically classified in [4] and [10] are examples
of tiling spaces to which the above would apply.
3. Spectral Information and Geometric Models. A detailed treatment of the spectral analysis of the natural flows on one-dimensional tiling
spaces is presented in [8]. The goal of this section is to indicate how these
results may be understood from our current perspective and how these results
can be used to construct geometric models of the tiling spaces in the sense
of [6]. In general, determining the discrete spectrum of a flow allows one to
determine a maximal semi-conjugate flow with pure discrete spectrum. In the
case of substitution tiling space flows, the substitution homeomorphism can be
modelled by the shift map on an inverse limit representation, [1].
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We explore this connection by examining the natural flows on tiling spaces
arising from substitutions of constant length on 2 letters. The spectral analysis
on the associated subshifts was carried out in [9]. As we shall see, when the flow
has pure point spectrum, not only is the flow measure theoretically conjugate
to a natural flow on an n-adic solenoid, but the shift map is also measure
theoretically conjugate to the shift map on the same n-adic solenoid.
Let σ be a substitution of constant length n on {a, b}: |σ (a)| = |σ (b)| = n :
na
nb
σ∼
,
n − na n − nb
where na is the number of a0 s in σ (a) and
similarly for nb . Then there is the
Z
substitution subshift (S, s) of {a, b} , s associated to σ, see, e.g., [2]. Let
T ∼ (L1 , L2 ) be the tiling space obtained from the special flow under function
f : S → (0, ∞) with
L1 , if x0 = a
f hxi ii∈Z =
.
L2 , if x0 = b
Then as in Corollary 1 there is the natural length-preserving map p : T → T 0
onto the associated space
f0
f1
K0 ← K1 ← K2 · · ·
T0 ∼ L
with the bonding maps fi the maps of the rose and with the factor spaces Ki
the wedge of 2 circles of lengths LM i .
With Si a circle of length L1 · ni , let Σ be the n -adic solenoid
g1
g2
S0 ← S1 ← S2 · · ·
Σ
where gi is a length-preserving n−fold covering. If L1 = L2 , then there is the
natural map q : T 0 → Σ which is induced by the mappings qi : Ki → Si which
“fold” the two circles in Ki onto the circle Si . The commutativity of the related
diagrams shows that the shift map on Σ is semi-conjugate to the substitution
homeomorphism of the original tiling space T . By the results of [9], q ◦ p is
a measure-theoretic isomorphism which is one-to-one off of a set of measure 0
in T .
Thus, Σ provides a model of both the flow on T and of the substitution
homeomorphism on T . Due to cohomological considerations, Σ cannot be
embedded in a surface, but the shift on Σ can be realized up to conjugacy
as the expanding attractor of a hyperbolic map on a solid (three-dimensional)
torus domain. The flow on Σ can also be realized up to conjugacy as a minimal
set of a flow on a solid torus domain, but this flow on Σ is uniformly Lyapunov
stable; whereas, the tiling space flow on T is not Lyapunov stable. Thus,
while Σ does provide a measure theoretic model of T , it does not share all the
significant dynamic properties.
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It follows from the results of [8] (see also [5]) that when L1 /L2 ∈
/ Q the
resulting tiling space flow on T is weakly mixing, and so there is no such
projection onto a solenoid, or even any periodic flow. Hence, the choice of L
makes a critical difference in the dynamics for this type of substitution.
The author thanks Lorenzo Sadun for very helpful conversations and advice.
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Received
December 10, 2002
University of Texas
Department of Mathematics
Denton, TX 76203-1430, USA
e-mail : alexc@unt.edu